AInstein: Numerical Einstein Metrics via Machine Learning

TL;DR

This paper introduces AInstein, a machine learning approach that solves Einstein's equations on manifolds without symmetry assumptions, providing insights into the existence of Ricci-flat metrics in higher dimensions.

Contribution

The paper presents a novel semi-supervised neural network architecture for solving Einstein's equations on complex manifolds without symmetry constraints.

Findings

Successfully solves Einstein's equations on spheres in dimensions 2-5.

Hints against the existence of Ricci-flat metrics on spheres in dimensions 4 and 5.

Provides a new computational tool for exploring geometric structures in differential geometry.

Abstract

A new semi-supervised machine learning package is introduced which successfully solves the Euclidean vacuum Einstein equations with a cosmological constant, without any symmetry assumptions. The model architecture contains subnetworks for each patch in the manifold-defining atlas. Each subnetwork predicts the components of a metric in its associated patch, with the relevant Einstein conditions of the form $R_{\mu \nu} - \lambda g_{\mu \nu} = 0$ being used as independent loss components (here $\mu,\nu = 1, 2, \cdots, n$, where $n$ is the dimension of the Riemannian manifold, and the Einstein constant $\lambda \in \{+1, 0, -1\}$). To ensure the consistency of the global structure of the manifold, another loss component is introduced across the patch subnetworks which enforces the coordinate transformation between the patches, $g' = J^T g J$, for an appropriate analytically known Jacobian $J$. We test our method for the case of spheres represented by a pair of patches in dimensions 2, 3, 4, and 5. In dimensions 2 and 3, the geometries have been fully classified. However, it is unknown whether a Ricci-flat metric can exist on spheres in dimensions 4 and 5. This work hints against the existence of such a metric.